^ Irvi 



ng Stringhan 





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M 



^- 



SHORT TABLE OF INTEGRALS 









COMPILED BY 



B. 0. PEIRCE 

HoLLis Professor of Mathematics and Natural Philosophy 
LN Harvard University 



BOSTON, U.S.A. 
GINN & COMPANY, PUBLISHERS 

1903. 






The compiler will he grateful to any person ivho may send 
notice of errors in these formulas to 

B. O. PEIRCE, 

Harvard College^ Cambridge. 



IN MEMORIAM 



•i U 



I. FUNDAMENTAL FORMS. 



ax. 



1. ladx = 

2. I af{x) dx=:a if(x) dx. 
Q rdx I 

+. I a;"*da; = , when m is different from — 1. 

J m -h 1 

5. \edx = e^. 

10. , _|^- .= yersin -'*. f ^ ■■ =• - cS«•^i^X ,. 

Ai» I COS a; aa; = sin a;. ^ 

/ 

I2» I Uin x dx = — cos x. 

86B425 



^ '; ..; ' 'FtridAMENTAL FORMS. 

15. ■rtana!seca;(to = seca;. 

16. j sec'' a; da; = tan a;. y^ 

17. )csc'a;da! = -rtna;. J 

I„ the following fo.™nlas. «, ., «-, -d y represent an, 
functions of X'. 

19a. Cudv=^uv-ydu, 



BATIONAIi ALGEBRAIC FUNCTIONS. 



II. RATIONAL ALGEBRAIC FUNCTIONS. 



A. — Expressions Involving {a-\-bx). 

The substitution of y or z for x, where yz=xz = a-^ fee, 
gives 

21. j\a-j-bx)'^dx = ^ Cy^'dy. 

22. (x{a-}- bx^dx = -^ fy^'iy — a)dy. 

23. (iir(a + bx)'^dx = -^^ i y"" {y — tiYdy. 

J x'^ia + bx)""" a'"+" V 2'" 

Wheuce 

26. C-^=.\\og(a-^bx), 
J a-\- bx b 

27. r ^^ = ^ 

J (a + fea;)2 6 (a + 6a;) 

28 r ^^ =: ^ 

' J (a + &a;)'* 26(a + 6a;)2 

29. r_£^ == 1 [a + 6a; - a log(a + bx)\ 
J a-\- bx b^ 

J {a-hbxy b'l ^^ ^^a + bx] 



D RATIONAL ALGEBRAIC FUNCTIONS. 

' J (a -h bxy b' [_ a + &a; 2(a + bxyj 

32. r_^^ = 1 [^(^ + 5a;)2 - 2a(a + bx) + cr log (a + 6a;)], 
J a -\-bx 0" 

S4. r_i^_=_iiog2^t*i^. 

J ic(a + 6a;) a a; 



35 



J a; (a + 50?)- a(a-f-5a;) a^ a; 

f__^^___ = _ J_ 4_ A log ^L±_^. 
J x^(a H- 6a?) ax a^ x 



36. 



rfa; 1 ^ la; 
tan ^ — 
c c 



^^ =_Llog^ + ^- 



B. — Expressions Involving (a H- 6a!f*) , 

38. I - 

' J c^ — a;^ 2c ^c — a; 

^^- f— 7T3 = -^ta'^^'«^\l^' if a>0, 6>0. 

*^- I , , 2 = .) / log-— -i: -^, if a > 0, 6 < 0. 

41, C—^^^—— — ^' L JL r rfa 

* J (a + 6a^^)^'~2a(a + 6a;2)"T-2aJ a + 6a;2* 

4i> r <^a; _ 1 X 2 m — 1 / * (?a? 

"' J (a + 6a:^)"*+i 27)ia (a + 6a;2)«"*" 2ma J(a + 6a2)' 

Ja + 6x-2 26 \ hj 



RATIONAL ALGEBRAIC FUNCTIONS. 

.. C xdx I C dz , - 

'*• I 7 , o, ^, = - I r-: — -,, where z = or. 

45, I = — log . 

J x{a + ba^) '2 a a + ba^ 

r oi?dx _ ^ _ « r dx 
' J a-\-bx- b hJa-\-ba? 

»/ it'^(a + 6.«-') ax aJ a-^-bx^ 

48. r '^c^^' _ —^ , I r dx 

J (a + 60^)"*+^ 2?rt6(a + 6a;2^"' 2m6J {a-^-ba?)"" 

J ay'ia + b3f)'^+' a J x^{a + bx^ a J (a + 6a;^)"'+i* 

where bk^ = a, 

51. r_^=XJiiog/^^lllMil^ + V3tan-^^^^'l 
J a-[-ba? ^bk\J \ {k-{-xY J^ h^l ] 

where bk^=a. 

52. f:._^ = -Llog-^. 

r da; ^1 r dx b r ^ 

' J {a-^bx'^y-^^ aJ(a-\-bx''y' aJ{a-\- 



x^'dx 
bx") 



'dx 
bx''y+^' 



• J (^a-{-bx^y+' bJ {a + bx'^y bJ (a + 6: 

r- r <^a; ^i rda; b T dx 

J ar^a + bx'^y^^ aJ x'^(a + 6a;'')p aJ xT ""(a + 6a;") 



IH-I 



RATIONAL ALGEBRAIC FUNCTIONS. 




II 

+ 

I 



'h 



I 

II 

a 

03 



II 




o 
A 

a 



> 



^ > 



^1^. 



o 
V 

?> 

c 
<v 



1 1 
> 

+ 



I ^ 

1 1 

> 

I 

+ 

I! § 

I ^ 
I I 
> 



^IX 



^|><l 



'tS 



'^ 



II 



BATIONAL ALGEBRAIC FUNCTIONS. 1 

64 r^dx __ _ 2a4-6g _ b(2n — 1) T^^ 

65. fA'do^^^-JLlogX + ^ll^l^^r^. 
J X c 2c' ^ 2c^ J X 

"^ J X^ cgX qJx 

' J A""+^ (2n-m + l)cX" 2n-m4-l'cJ X"+^ 

'dx 



■ m — 1 ^ a r^ 
2n — m-f-1 cJ X 



n+l 



•Ja;X 2a ^X 2aJ X 

rdx ^ b , X L4./'_^_^^^^ 

'Ja;-X 2a2 ^a^ aa; \2a2 aJJX 

70 r__^^_ = \ ??. 4-^1 — 1 6 r dx 

• J aj'^X^+i (m - l)ait"»-iX»' m - 1 * a J x'^'^X' 

2n-\-m — 1 c^ r dx 
aJ a 



D. — Rational Fractions. 

Every proper fraction can be represented by the general 
form : 

/W ^ 9i ^""'-^ 92 ^''~' + 93 X''-' + - + f7„ 
F{x) x^ + k^x"" ^ -{- ksx""-'^ -i \-k,, 

a, 6, c, etc., are the roots of the equation F{x)=0, so 
that 

F(x) = (X- - ay (a; - 6)« (a; - cr •-, 



10 RATIONAL ALGEBRAIC FUNCTIONS. 

then /W_ = ^4^ + ^^^— + A_+...+-A_ 

F {x) {x - ay {X- ay ' {x - ay'' x-a 

f— ^i— + ^ + ^ 4--+-^ 

(x- by {x- by-' (x - by-' x-b 

I ^1 _| Yl J ^3 _L. ... _J_ ^r _ 

{x — cy (x — cy-^ {x — cy^ x—c 



Where the numerators of the separate fractious may be 
determined by the equations 

,^ (.) =«g^^ <^. (-) =^^^J^^ etc., etc. 
If a, 6, c, etc., are single roots, then p = g = ?•=••• = 1, 

""•^ /(a--) _ ^ , J? , c 
P" (a?) a; — aa; — 6a; — c 

where vl — -^ ^^^ B — ^^^' eto 

where -^ - p, ^^y ^ - p, ^py ^^- 

The simpler fractions, into which tjhe original fraction is 
thus divided, may be integrated by means of the following 
formulas : 

r hdx _ rhd(mx-\-n) 7i 

J {mx-\-ny J m {mx -\- ny m {1 — I) {mx -}- ny~^ 

C hdx h I X , >, 
72. I == — log (mx + n) . 

J mx-^n m 

If any of the roots of the equation f{x) =0 are imaginary, 
the parts, of the integral which arise from conjugate roots 
can be combined together and the integral brought into a 
real form. The following formula, in which i=:V— 1, is 
often useful in combining logarithms of conjugate complex 
quantities : 

78. log {X ± yi) = i log (x2 + /) ± i tan ' •^. 



IKBATIONAL ALGEBRAIC FUNCTIONS. 11 



III. IRRATIONAL ALGEBRAIC FUNCTIONS. 



A. — Expressions Involving Va -|- hx. 

The substitution ,of <a new variable of integration, 
y = Va + bx^ gives 

74. ( Va ■+■ hx dx = ^ ^/{a + bxy. 

75. r«V^r+ted.- = -^i?iLz^5MVIH+M!. 

J 1056^ 

J X ^ ^yja-j-bx 

r dx _ 2 V ft 4- 6a? ^ 

C ^d^ '2(2a — bx) / . , , 

79. I . = '^-TTi ^VA + 6a;. 

80. f >-^^^ ^2(8a^-4a6.. + 36^a^)^^^-^:^ 

81. r_^g^ = -JLlog/: -^'- f \ for a>0. 
•^^ a' Va + 6a; Va VVa.+ 6a; + Va/ 

^^ r da; 2 . _ i la + ^^' i?^ ^ ^ rk \ 

S2. I =: = tan ^^ — ' , for a < 0. ^ 

Ja;Vrt + 6a; V — a ^ — « 

r (Jx _ V a + 6a; 6 r dx 
•^a^Vo^^" <*^ • -«^ -^Va + da; 



32 



IRRATIONAL ALGEBRAIC FUNCTIONS. 



84. J{a + bx)^'idx = ^Jy'^-dy = ?-^ 



hx) 



(2 ±71) 



85. \ x(a-\-hx)~^dx = —\ ^ — ■ ^ ^^ — ' '- — . 

J ^ ' W\_ 4.±7l 2 ±71 J 

J ^ x'^dx _ 2x"' Va H - hx 2 7?ict /* x'^-'^dx 

^'a + hx~ (27/i+l)& (27?iH-l)6J VoTfe^ 

/ da; _ _ Va + 5a7 _ (2??. — 3) & r da? 

a'" Va + 6a; 0^-1) «^'" ^ (27i - 2) a J a^'^-VoH^ 

88. f " + ^;)^ ^^ = 6 J(a 4- hx)'^d. + a f ^^ "^f ^"^^ ^^^ 



86 



87 



89./- 



da; 



= 1 C clx -k C- 



dx 



X (a + bx) 2 '^'" a; (a + 6a;) 2 ^""^ (a + 6ic) 2 

B. —Expressions* Involving Va.-^ ± a^ and \la-—x^, 

90. \^x^± a' dx = i \_x Va;' ± «^ ± a- log (a; + Va;''^±a^) ] . 

91. I Va'— ar da; = -J-[a;Va'—a;^ 4- a- sin"^-]* 



92 



•J; 



dx 



^x'±a? 
dx 



log( 






— cir»~l 



94 
95 



1 !« I 

-COS ^-» — - -i- 



Ja;Va2±a;2 a \ a; / jx/iVx- «t »- 

I - — = — dx = Va- ± ar — a log ^ \ l=^r^ — ^ 



Va^ — x^ 

dx _ 

= -eos -— r=: ~- iec :^ 

a;Va;2_^2 a a; O- o^ 



•These equations are all special cases of more general equations given in the next sectioa 



97 



IRRATIONAL ALGEBRAIC FUNCTIONS. 13 



J X X 

98. f . ^^^ = ± -J'oF±^, ' 

100. fx Va^ ± o? dx = 1 V(ar^±a2)S. 

101. I ic Va"'' — af' dx- = — J V(ct- — a;^)^ 

102. Cy/(x' ± crydx 

= i\ x^ix' ± d'y± ^^\/¥±d' + ^\og{x +y/W±d')^ 

103. C'\/{a' — x-ydx 

if /7~^ — i:^Vs I 3a*.r / -i r, , oa^ . _ix"l 

= ^ ajV(a- — ar)^ + - va-— :r4-- sin^ . 

[_ 2 2 aj 

104. r ''^ ±" 



105, 



da; _ a; 

V (a- - ar'p ~ «' -Ja'-a^ 

xdx — 1 



f— ^ 

^ -slid' 

106. f , , _ 

107. r--:^^=.=— i=. 

108. rxV(ar^±aO«da; = J V(^^^^ 

109. CxyJid'-x'Ydx^^ -\\l{a- — xy. 



14 IRRATIONAL ALGEBRAIC FUNCTIONS. 

110. I x^^/af± a^dx 

= -y/{x'±ay If - {x\/¥±^' ± a2 log (x + V^T^)), 
4 8 

111. Ix^y/a^ — x^dx 

112. f ^^^ = ^' V^^±^ ^F ^'log (X -t- V^db^n. 

113. r ^^^ ^_gv,,2i:^+gL%ia-i^. 



2 2 a 



118 



114. I =± ; 

115. f ^^ =-:Zg^. 

J a- a; 

J ay^ X a 

r x^dx _ ^ -X 4_iog(a;-l-V^±^V 

119. f ^^^ = ^ -sin-^^. 

C. — Expressions Involving Va -\-hx + cx^. 

Let X = a + 6a; + ca*^, g = 4 ac — 6", and h=z~ In order 

to rationalize the function /(a;, yl a + hx -\- ca?) we may put 
Va ■\-hx-\-cx^= yl ±c\l A+Bx ± a;^, according as c is positive 
or negative, and then substitute for x a new variable 2, such 
that 



IRRATIONAL ALGEBRAIC FUNCTIONS. 15 



z = V^ -^Bx -h ar — x, if r- > 0. 






where a and yS are the roots of the equation 

^+jBic-a^2^0, if c<0 and -^<0. 

— c 

By rationalization, or by the aid of reduction formulas, may 
be obtained the values of the following integrals : 

20. f-^ = J^iogfVX 4- oj Vc + -^\ if c> 0. 
^ -^X, Vc V 2Vcy 

a-i r da; 1 . _,/— 2ca;~6\ ... ^^ 

21. I-— ==-— =sin M . if c<0. 

22. r ^«^ ^ 2(2ca; + &) , 

23. f_J^^2(2ca. + 6)/l_^^A 

^ X''\fX 2>q\IX \X J 

24. r dx ^ 2(2cx-\-b^/X 2k(n — l) r dx 

J yr-^nt (27i-l)oX^ 2n-l J irn-iJY-' 



X«VX (2n-l)gX" 27i-l J X^-'^X 

25. C^Xdx = i^^^±^l^ + ± f— - 
J 4c 2kJ ^x 

26. fxVXda. = (2-dLaVX/^ AA + 3 T^. 

J 12c V 4:k^8ky^lwJ sjx 

28. rx-vxda;= (^^^+^>^^^ + ^^+^ r^^. 

J 4(n + l)c 2(n + l)kJ VX 

'-^ ^/x'" c 2cJ Vx' 



16 IRRATIONAL ALGEBRAIC FUNCTIONS. 

130. C ^^^^ = '^(b^-h'2a) 

-^ X" VX (2n- l)cX" 2cJ X"Vx' 



r QiTdx ^ {2b' — 4:ac)x-{-2ab 1 f 



dx 



r x-dx ^ (26^-4ac)a;+2a6 4ftc+(2yi-8) 6^ T c 

'-'X'^VX (2w-l)c^X»-WX (2n-l)cg Jx- v .v 

^^v vx~l^~r2'c^ 8^"3^y ■^U'^~T^P vx' 

1 36. faj VXdoj = ^^^ - - C-s/Xdx. 
J ■ 3 c '2 c J 

1B7. (xX VXdo^ = ^^^ - ^ fxVXcia;. 
J DC 2cJ 

'•^ VX {2n + l)c 2cJ Vx * 

J \ 6cJ 4c 16c^ J 

V VX 2(n + l)c 4(7i + f)cJ VX 

a r X^'dx 

2(n4-l)cJ VX * 

141. Ca^y'Xdx=(x--l^^^-^-^'\^:^. 
J \ 8c 48c2 ScJ DC 



IRRATIONAL ALGEBRAIC FUNCTIONS. 17 

.„ r c^a; 1 . ^f bx-\-2a \ .J. ^ 

.43. I — — = --=sin U — ^ , if a<0. 

^a;VX V-a \xyJb'^-AacJ 

^ x^JX bx 

45. r_^i__= Vx ^ 1 r dx b r dx 

/ ' dx _ \/X b_ r dx 
^-</ IT ax 2aJ r. 



a^Vx ^^ '^^^ x\/X 

dx 



dx 



J X 2J yjx ^ X 

'J xyJX (2n-l)VX -^ a?yX 2 J VX 

'•^ ^' ^ 2-^a;VX ^^WX* 

r_^cZa;_ ^ 1 r x^-^dx _ b Px"" ^dx _ a raf'^dx 
'-'xWX c J x«-WX cJx«VX C'^ x»Vx* 

ra; "'X"da; ^ a^'^-^X^^VX (2n + 2m- 1) 6 r a7'"-^Y"(Za; 
'•^ VX (27i + m)c 2c(27i + m) J ^x 

(m— l)rt r o;"* -X"da; 
(2?i + 7?i)cJ VX 

52 f ^^ _ ^ V^ 

V arX»VX (m-l)aaf"-^X« 

(2yi+2m-3)6 ^ da; ( 2?i+m--2)c r da; 

2a(m — 1) ^ x'"-^X'"\/'X im — \)a J x"'~^X*'\fX 



18 



IRRATIONAL ALGEBRAIC FUNCTIONS. 



153 fX^'dx ^ X^-'VX (2n-l)b rX^-'dx 
'Jx""\/X (m — 1)0;"'-^ 2(m — 1) J aj'^-^VX 



(2?i-l)c rX^'dx 

1 J ^-2^X 



m 



54. rv2 



D. — Miscellaneous Expressions. 



2 



a 



versin"^— 

^2ax — x^ « 



*^ V2ax — „ 

56. f ^^^ = + J^. 

57. f ^^_= = _ J^S. 

J (a; _ 1)7.^2-1 \a;-l 

68. J Ji±5c?a; == sin-la; - VT=^- 

™ X — X 



+ (a - 6) log ( Va; + a + Vo; + 6). 



^ ^{x-a)( 



2 Sin \ . 

\i8-a 



(fi-x) ^^ 

61. f ^ = _l,sin-\^i^±M, 

62. r-v/a + 6.rda; = — ^/(a + &a?)*. 
^„ r dx 3 8/^ — , , .^ 



IRRATIONAL ALGEBRAIC FUNCTIONS. 19 



164. r-,-gg^^- ^^^^-f^^) ^(^Tw. 

165. I — = — sec V— • 




20 J TBANSCENDENTAL FUNCTIONS. 



IV. TRANSCENDENTAL FUNCTIONS. 



cos X. 



07. i sina^'dx* 

G9. I sin'^it' da; = — | cos x (sin^a; + 2) . 

<<). I sm"a:f/.r = 1 I sin" ^xdx. 

J n n J 

71. I cos x dx = sin x. 

72. i Qos-xdx=i^^mxQo^x-\-^x. 

73. j cos^fl7dcc = |^siu.T?(cos-a;-|- 2). 

74. I cos'*ii'c/a;= cos" ^:); sin.x'-j —- I cos" ^xdx. 

75. I sin X cos x dx = ^ si n- ic . 

70. I sin^ii" iioa-xdx = — g (^sin 4a; — x) . 



7.Jsi 



77. i sin X cos"* a; fZa; = — 



COS'"+^T 

m + 1 



78. I sin"'a;cosa;cfa; 



I sin'" a 



S^' 



79. I cos"' a; sin" a; da; 



m+l 

cos"*-^^a; sin""^'a; 



m 4- ?i 

_^m-^ (cos"" 2 a; sin" a; da, 
m 4- »<^*^ 



«QA r m • n 7 sin'*~^^• cos" 

180. I cos"* a; sin" a; da; = 

J m + n 



'^— — - j cos'"a;sin'*"^a!;daj 



TRANSCENDENTAL FUNCTIONS. 21 



181 r<^'OS'"a:dU-_ c os"' "^ 07 m — n + 2 /' cos'" 

.7 sin"a; (?i — l)sin'*'^a/* 9i — 1 %/ sin" ^oj 

jg2 rc os'^xdx _ cos" *"^ X , m — 1 r cos"'~^xdx 

J sin" a; (??i — ?i)sin" '.r m — nJ sin" a; 

cos"Y''~a;Vr---^'^ 

*/ sin*;* cc cos" it* 

1 1 . m 4- n — 2 /^ dx 



1 m + ^ ? — 2 r ( 

"^aj.cos"^a; ?<, — 1 •/ sin'"a;. 



71—1 sin"* ^aj.cos"^a; ?<, — 1 •/ sin'" ic. cos" ^x 

1 1 7?i + 72 -5- 2 /^ dx 



72 -r- 2 r 

— 1 J sin*" 



85 



771 — 1 sin"*"^ X . cos" ^^• 7?i — 1 J sin'"^^it' . cos" a; 

dx 



J ^ dx _ 1 coso; m — 2 /^_ 

sin'" a; m — 1 sin"''\T 7)i — iJ sin'" "a; 

' C ^^ — 1 sin a; n — 2 / * da; 

*./ cos"a; 71—1 cos"^a; n — iJ cos"'~^x 

87 . I tan xdx = — log cos x. 

88. I tan^ajda; = tan x — x. 

89. rtan"a;da; = ^^^H^^-^- ftan" 2a;c?a5. 
J 71 - 1 J 

90. j ctna;c?a; = logsina!. 

91. I ctn^a;da;= — etna; — a;. 

92. rctn"a;da; = - ^^^^" ^ -- fctn" ^ajda?. 

93. j seca;da; = log tan f j + - J» 

94. j sec^a;da; = tana;. 



22 TRANSCENDENTAL FUNCTIONa 

195. Csec^xdx= f-^. 
J J cos"* a? 

196. j csca;da; = log tanja;. 

197. i c&c^xdx = — ctnx, 

198. I csc'*fl?da;= I — 

'Ja + b cosa; ^a^^i^^ \_a-i-b cosccj J I 4 ««y >^ j c#» 
'6 + a cosa.' + V6^ — a^ . sma;~| =^fSn -^ 

200. f— ^-^ 



1 , r 6 + g cos a.' + Vfe'^ — a^ . sma; ~[ 
' !_^2 [^ a + 6 cos a; • J 



+ c sin a? 

— 1 . _ir 6^4-c^ + <^(^ cosa;H-c sina?) n 

Va^ — W — c^ LV6^+ c^ (a + & cosa; + c sina;) J 

1 



• log; 



V6' 4- c2 - a' 

[ 5^ + c^ + g (& cosa? + c sina;) + V6^ -\- (^ — a^ (b sin a; — c cos a;) "] 
V6^ + c^(aH-6cosa; + csina;) J 

201. j a; sin a; da; = sin a; — a; cos aj. 

202. I a^sina;cfa; = 2 a; sin a; — (a;^ — 2) cosa?. 

203. j x^ sin a;da; = {Sx^ — 6) sin a; — (a;^ ~ 6 a;) cosa;. 

204. j af" sin a;da; = — a;"* cos a; + m j a;"*"* cosajda;. 
206. I X cos a; da; = cos x-j-x sin a?. 

206. j ar^cosa;da;= 2a;cosa; + (a;^— 2) sina;. 

207. j a;^ cos a; (Za; = (3 a;^— 6) cosa;-h(a;^ — Oa;) sinaj. 



TRANSCENDENTAL FUNCTIONS. 



208. I x"" cos X clx = x"' s\n X — m I ic"*"^ sin x dx. 

209. f^lBI clx = L_ . !lE^ ^ _i_ f^2^ ax, 

J x"" 711 — 1 a;"* ^ m — 1 J a;'"-^ 

210. CS2^dx= ^— . 2^^ L_ C^^^dx. 

J of' m — 1 a?*" ^ m — iJ aj"*"^ 

211. C^J^dx = x--^ + -^--^ + -^.,., 
J X 3.3! 5.5! 7.7! 9.9! 

-^- Cao^x-i ■, x^ , X* x^ , x^ 

212. I dx = \oa^x ..» 

J X ^ 2.2! 4.4! 6.6! 8.8! 

aid. I sin ma; smna/*aa;= — ^^ ^ — >^ — — — ^—, 

J 2{m^n) '2{m-\-n) 

214. Ccosmxcosnxdx= ^'"<^ " ""> + "'"^^^ + ">'" ■ 
J 2{m — n) 2{m + n) 

215. I Bm'^xdx== a; sin ^ a; H- V 1 —x^. 
216.^ I cos ^ a; da; = x cos^^a; — V 1 — x^, 

217. j tan^a;da; = a;tan^^a; — ^log(l +a;^). 

218. I ctn~^a;da; = a;ctn~^a;4- Jlog(l -f- a;'0* 

219. I versin~^a;da;= (a;— 1) versin'^a; + -sj'lx — x^^ 

220. j (sin~^a;)2da: = a;(sin"^a;)^ — 2a;4-2Vl — a;^sin"*aj. 

221. j a;.sin"'^a:da; = :^[(2a.*^— 1) sin^^a; + a;Vl — a;"^]. 

««^ r » • -1 7 aj^+^sin-^a; 1 Cx^'^^dx 

222. I aj'^sin ^xdx= I 

J n-\-l ?i + 1 . ' V I - Jr 

«^.» Cn -1 ^ a;"+^cos"^a; , 1 /".f"+*da; 

223. I X" cos^ a; da; = 1 I — » 

./ w + l w + U Vl — a;^ 

«^. C n, I ^ a;"+Han ^^' 1 fa^^+i* 

224. Ia''*tan ^xdx — I 

J n-\-\ n + lJ H- 



dx 
a?' 



24 TRANSCEKDENTAL FUNCTIONS. 

225. j log X dx = X log X — .'>;. 

220. ril2g^^c^.:._JL(iog.^)«.i. 

227, I 

t/ .T logo; 

228. (- 
J X 



dx , 

= log, logic. 



dx 



{\ogxY (7i-l)(loga^)"-i 

229. i a;™ log ic f/a; = o;-^ i Tl^^ ^— 1 ( ^»* L ( * « ) - ^'*' k 

230. re"-f?a; = — . i_ [^^ 

231. Cxe'''dx==^(ax-1). 

232. ra;-e-d^ = £l!£!-.!'^. fo^-^e-t^a^. 
*^ a aJ 

233. r^d^ = 1_^+_JL_ C_^a^ 

J xr m-la.— i^m-lJx- 1 

234. fe^^ Xo^xdx = ^!!i2S^ _ 1 f^!! ,a- 
^^ a aJ X 

235. f V-' ^.-n o^^.. - ^"" (« sina^ - coso;^ 

^ a^ + l • 

236. fe-cos:crfa; = £l(^LS£i^±_SH^. 



DEFINITE INTEGRALS. 25 

DEFINITE INTEGRALS. 



2.S7. f -^^_ = '', if a>0; 0, if a = 0; -'', if a < 0. 

Jo cr + x- 2 2 

238. j ic"-ie~''da?= j log- dx = T{n). 

T{n+l) = n'T{7i). r(2) = r(l)=l. 

r (?i -f- 1) = w !, if n is an integer T {^)— ^/tt, 

Jo ^ Jo (l-fa;)'"+" r(m + w) 

IT IT 

240. j sin''a;da;= | cos'^xdx 
Jo Jo 

s=__L_: — "•K'^^— — ; . ![ jf jj ig ajj gygjj integer. 
2.4.6. ..(w) 2 

_ 2.4.6...(yi-l) .^ ^^ jg ^^ ^^^ integer. 
1.3.5. 7. ..ii 

— i /^ V / , for any value of n, 

r-sinmxdx^^^ if m>0; 0,ifm = 0; - |, if m < 0. 

Jo a; 2 - 

r-smx.eosmxdx^^^ if m<-l or m > 1 ; 

Jo X 

^, if m = -l or m=l; ^, if -l<m<l. 
4 ^ 



241, 
242, 



2^g r sirrxd x _'7r 

\h x" ~2 

244. r* cos (a^) da; = f* sin (a;^) fia; = i J|. 



26 DEFINITE INTEGRALS. 

Jo l-{-x' 2 
2-« ■ r'*' Gosxdx _ r°^ sinxdx _ B 

Jo ^x ^0 V^ \2 



247. ' ^^ 






Vl — A;'-^ sin^a; 



=l[i + (i)^^+6:!>'+ (lxlj^^+ ••■} « ^< 



= /r. 



248.J^ -s/l-k^sin^x.dx 

■.i[:-<«-*--(HJf-(l±5)-f-...}„^< 



249. 



Jo 2a^ 2a ^^ 

Jo a"+^ a''+^ 

Jo 2"+^ a" \a 

252. r.-2c?.=^:!:^. 

Jo 2 

253. f e-«*cosma;da; = — -^ — -, if a>D. 
Jo a^ -f- m^ 

254. I e'''^8m'mxdx= ■ ;,, if a>0. 

Jo a- + 7R- 



ft2 



255. I e-°'''^cos6a;da;= V^-^ ""' . 



2a 



256. fMf^d^ = -ZL' 
Jo 1 - a? 6 

267. fM^d^ = -^. 
Jo l-{-a? 12 



DEFINITE INTEGRALS. 27 

268. r^^dx = -^. 
Jo 1—x^ 8 

259. riog/^i±^V- = ^. 
Jo \l-xj X 4 

261. f '^^ =./i. 

262. rVlogAY,^^=^(" + 0. 
Jo =\^a;y (TO + !)»+' 

263. I logsina!da!= | logcosac?a; = — - • log2. 

ic . log sin oj da; = — — log 2. 



28 AUXILIARY FOKMULAS. 

AUXILIARY FORMULAS. 



The following formulas are sometimes useful in the reduction 
of integrals : 

266. logu = \ogcu -\- a constant. 

266. log(— ^^) = logu -f a constant. 

- sinWl —u^ + a constant. 

267. sin^^w = ■{ —^sin~^{2u^— 1) +a constant. 
^sin~^2i* Vl —u^ + a constant. 

> — tan^ + a constant. 

268. tan-^?«=^ ^ 

tan"^ — — h a constant. 

L 1 — cu 

269. log {x ± yi) = ^log {ay^ + ?/) ± itan"^--- 

X 

270. sin~^t^ = cos~Wl — u^ = tan^ = csc~^ — 

Vl - u- '" 



il 
= sec^^ - 



271. cos~^?* = sin Vl— ti^ = tan ^\ — 

272. tan^^aj ± tan^2/ = tan V-^^ — ^V 

Vl q: xyj 

273. sin~*ic ± sin" ^2/ = sin^^ (xy/l —y^±yy/l —x-). 

274. cos~^ X ± cos~^y = cos ~^ (xy ^: y/{l — x^){l —y^)). 

275. sma;= --— 

2i 

e'* 4- e~*' 

276. cosa;= '- 

277. sina:/ =^i(e'' — e"*)= isinha;. 

278. cos xi = ^ (e^ + g-^^) = cosh x, 

279. log,;r = (2.3025851) logioa;. 



TABLES. 



29 



The Natural Logarithms of Numbers between 1.0 and 9.9. 



N. 


O 1 

! 


2 


3 


4 


5 


6 


7 


8 


9 


1. 


0.000 


0.095 


0.182 


0.262 


0.336 


0.405 


0.470 


0.531 


0.588 


0.642 


2. 


0.693 


0.742 


0.788 


0.833 


0.875 


0.916 


0.956 


0.993 


1.030 


1.065 


3. 


1.099 


1.131 


1.163 


1.194 


1.224 


1.253 


1.281 


1.308 


1.335 


1.361 


4. 


1.386 


1.411 


1.435 


1.459 


1.482 


1.504 


1.526 


1.548 


1.569 


1.589 


6. 


1.609 


1.629 


1.649 


1.668 


1.686 


1.705 


1.723 


1.740 


1.758 


1.775 


6. 


1.792 


1.808 


1.825 


1.841 


1.856 


1.872 


1.887 


1.902 


1.917 


1.932 


7. 


1.946 


1.960 


1.974 


1.988 


2.001 


2.015 


2.028 


2.041 


2.054 


2.067 


8. 


2.079 


2.092 


2.104 


2.116 


2.128 


2.140 


2.152 


2.163 


2.175 


2.186 


9. 


2.197 


2.208 


2.219 


2.230 


2.241 


2.251 


2.262 


2.272 


2.282 


2.293 



The Natural Logarithms of Whole Numbers from 10 to 109. 



N. 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


1 


2.303 


2.398 


2.485 


2.565 


2.639 


2.708 


2.773 


2.833 


2.890 


2.944 


2 


2.996 


3.045 


3.091 


3.135 


3.178 


3.219 


3.258 


3.296 


3.332 


3.367 


3 


3.401 


3.434 


3.466 


3.497 


3.526 


3.555 


3.584 


3.611 


3.638 


3.664 


4 


3.689 


3.714 


3.738 


3.761 


3.784 


3.807 


3.829 


3.850 


3.871 


3.892 


5 


3.912 


3.932 


3.951 


3.970 


3.989 


4.007 


4.025 


4.043 


4.060 


4.078 


6 


4.094 


4.111 


4.127 


4.143 


4.159 


4.174 


4.190 


4.205 


4.220 


4.234 


7 


4.248 


4.263 


4.277 


4.290 


4.304 


4.317 


4.331 


4.344 


4.357 


4.369 


8 


4.382 


4.394 


4.407 


4.419 


4.431 


4.443 


4.454 


4.466 


4.477 


4.489 


9 


4.500 


4.511 


4.522 


4.533 


4.543 


4.554 


4.564 


4.575 


4.585 


4.595 


10 


4.605 


4.615 


4.625 


4.635 


4.644 


4.654 


4.663 


4.673 


4.682 


4.691 



The Values in Circular Measure of Angles which are given In 
Degrees and Minutes. 



1' 


0.0003 


9' 


0.0026 


3° 


0.0524 


20° 


0.3491 


100° 


1.7453 


V 


0.0006 


10' 


0.0029 


40 


0.0698 


30° 


0.5236 


110° 


1.9199 


V 


0.0009 


20' 


0.0058 


5° 


0.0873 


40° 


0.6981 


120° 


2.0944 


4' 


0.0012 


30' 


0.0087 


(P 


0.1047 


50° 


0.8727 


130° 


2.2689 


5' 


0.0015 


40' 


0.01 16 


70 


0.1222 


60° 


1.0472 


140° 


2.4435 


6' 


0.0017 


50' 


0.0145 


8° 


0.1396 


70° 


1.2217 


150° 


2.6180 


V 


0.0020 


1° 


0.0175 


90 


0.1571 


80° 


1.3963 


160° 


2.7925 


8' 


0.0023 


2° 


0.0349 


10° 


0.1745 


90° 


1.5708 


170° 


2.9671 



30 



TABLES. 



NATURAL TRIGONOMETRIC FUNCTIONS. 



Angle. 


Sin. 


Csc. 


Tan. 


Ctn. 


Sec. 


Cos. 




0° 


0.000 


00 


0.000 


00 


1.000 


1.000 


90° 


1 


0.017 


57.30 


0.017 


57.29 


1.000 


1.000 


89 


2 


0.035 


28.65 


0.035 


28.64 


1.001 


0.999 


88 


3 


0.052 


19.11 


0.052 


19.08 


1.001 


0.999 


87 


4 


0.070 


14.34 


0.070 


14.30 


1.002 


0.998 


86 


5° 


0.087 


11.47 


0.087 


11.43 


1.004 


0.996 


85° 


6 


0.105 


9.567 


0.105 


9.514 


1.006 


0.995 


84 


7 


0.122 


8.206 


0.123 


8.144 


1.008 


0.993 


83 


8 


0.139 


7.185 


0.141 


7.115 


1.010 


0.990 


82 


9 


0.156 


6.392 


0.158 


6.314 


1.012 


0.988 


81 


10° 


0.174 


5.759 


0.176 


5.671 


1.015 


0.985 


80° 


11 


0.191 


5.241 


0.194 


5.145 


1.019 


0.982 


1 79 


12 


0.208 


4.810 


0.213 


4.705 


1.022 


0.978 


78 


13 


0.225 


4.445 


0.231 


4.331 


1.026 


0.974 


77 


14 


0.242 


4.134 


0.249 


4.011 


1.031 


0.970 


76 


15° 


0.259 


3.864 


0.268 


3.732 


1.035 


0.966 


75° 


16 


0.276 


3.628 


0.287 


3.487 


1.040 


0.961 


74 


17 


0.292 


3.420 


0.306 


3.271 


1.046 


0.956 


73 


18 


0.309 


3.236 


0.325 


3.078 


1.051 


0.951 


72 


19 


0.326 


3.072 


0.344 


2.904 


1.058 


0.946 


71 


20° 


0.342 


2.924 


0.364 


2.747 


1.064 


0.940 


70° 


21 


0.358 


2.790 


0.384 


2.605 


1.071 


0.934 


69 


22 


0.375 


*2.669 


0.404 


2.475 


1.079 


0.927 


68 


23 


0.391 


2.559 


0.424 


2.356 


1.086 


0.921 


67 


24 


0.407 


2.459 


0.445 


2.246 


1.095 


0.914 


66 


25- 


0.423 


2.366 


0.466 


2.145 


1.103 


0.906 


65° 


26 


0.438 


2.281 


0.488 


2.050 


1.113 


0.899 


64 


27 


0.454 


2.203 


0.530 


1.963 


1.122 


0.891 


63 


28 


0.469 


2.130 


0.532 


1.881 


1.133 


0.883 


62 


29 


0.485 


2.063 


0.554 


1.804 


1.143 


0.875 


61 


30° 


0.500 


2.000 


0.577 


1.732 


1.155 


0.866 


60° 


31 


0.515 


1.942 


0.601 


1.664 


1.167 


0.857 


59 


32 


0.530 


1.887 


0.625 


1.600 


1.179 


0.848 


58 


33 


0.545 


1.836 


0.649 


1.54Q 


1.192 


0.839 


57 


34 


0.559 


1.788 


0.675 


1.483 


1.206 


0.829 


56 


35° 


0.574 


1.743 


0.700 


1.428 


1.221 


0.819 


55° 


36 


0.588 


1.701 


0.727 


1.376 


1.236 


0.809 


54 


37 


0.602 


1.662 


0.754 


1.327 


1.252 


0.799 


53 


38 


0.616 


1.624 


0.781 


1.280 


1.269 


0.788 


52 


39 


0.629 


1.589 


0.810 


1.235 


1.287 


0.777 


51 


40° 


0.643 


1.556 


0.839 


1.192 


1.305 


0.766 


50° 


41 


0656 


1.524 


0.869 


1.150 


1.325 


0.755 


49 


42 


0.669 


1.494 


0.900 


1.111 


1.346 


0.743 


48 


43 


0.682 


1.466 


0.933 


1.072 


1.367 


0.731 


47 


44 


0.695 


1.440 


0.966 


1.036 


1.390 


0.719 


46 


45° 


0.707 


1.414 


1.000 


1.000 


1.414 


0.707 


45° 




Cos. 


Sec. 


Ctn. 


Tan. 


Oec. 


Sin. 


Anglo. 



TABLES. 



31 



Values of the Complete Elliptic Integrals, K and E, for Different 
Values of the Modulus, k. 



sin-iifc 


K 


E 


sin-iA; 


K 


E 


sin-U- 


K 


E 


0° 


1.5708 


1.5708 


30^^ 


].6858 


1.4675 


60° 


2.1565 


1.2111 


1° 


1.5709 


1.5707 


31° 


1.6941 


].4608 


61° 


2.1842 


1.2015 


2° 


1.5713 


1.5703 


32° 


1.7028 


1.4539 


62° 


2.2132 


1.1920 


3° 


1.5719 


1.5697 


33° 


1.7119 


1.4469 


63° 


2.2435 


1.1826 


40 


1.5727 


1.5689 


34° 


1.7214 


1.4397 


64° 


2.2754 


1.1732 


50 


1.5738 


1.5678 


35° 


1.7312 


1.4223 


65° 


2.3088 


1.1638 


6° 


1.5711 


1.5665 


36° 


1.7415 


1.4248 


66° 


2.3439 


1.1545 


70 


1.5767 


1.5649 


37° 


1.7522 


1.4171 


67° 


2.3809 


1.1453 


8° 


1.5785 


1.5632 


38° 


1.7633 


1.4092 


68° 


2.4198 


1.1362 


90 


1.5805 


1.5611 


39° 


1.7748 


1.4013 


69° 


2.4610 


1.1272 


10° 


1.5828 


1.5589 


40° 


1.7868 


1.3931 


70° 


2.5046 


1.1184 


11° 


1.5854 


1.5564 


41° 


1.7992 


1.3849 


71° 


2.5507 


1.1096 


12° 


1.5882 


1.5537 


42° 


1.8122 


1.3765 


72° 


2.5998 


1.1011 


13° 


1.5913 


1.5507 


43° 


1.8256 


1.3680 


73° 


2.6521 


1.0927 


14° 


1.5946 


1.5476 


44° 


1.8396 


1.3594 


74° 


2.7081 


1.0844 


15° 


1.5981 


1.5442 


45° 


1.8541 


1.3506 


75° 


2.7681 


1.0764 


16° 


1.6020 


1.5405 


46° 


1.8691 


1.3418 


76° 


2.8327 


1.0686 


17° 


1.6061 


1.5367 


47° 


1.8848 


1.3329 


77° 


2.9026 


1.0611 


18° 


1.6105 


1.5326 


48° 


1.9011 


1.3238 


78° 


2.9786 


1.0538 


19° 


1.6151 


1.5283 


49° 


1.9180 


1.3147 


79° 


3.0617 


1.0468 


20° 


1.6200 


1.5238 


50° 


1.9356 


1.3055 


80° 


3.1534 


1.0401 


21° 


1.6252 


1.5191 


51° 


1.9539 


1.2963 


81° 


3.2553 


1.0338 


22° 


1.6307 


1.5141 


52° 


1.9729 


1.2870 


82° 


3.3699 


1.0278 


23° 


1.6365 


1.5090 


53° 


1.9927 


1.2776 


83° 


3.5004 


1.0223 


24° 


1.6426 


1.5037 


54° 


2.0133 


1.2681 


84° 


3.6519 


1.0172 


25° 


1.6490 


].4981 


55° 


2.0347 


1.2587 


85° 


3.8317 


1.0127 


26° 


1.6557 


1.4924 


56° 


2.0571 


1.2492 


86° 


4.0528 


1.0086 


27° 


1.6627 


1.4864 


57° 


2.0804 


1.2397 


87° 


4.3387 


1.0053 


28° 


1.6701 


1.4803 


58° 


2.1047 


1.2301 


88° 


4.7427 ! 1.0026 


29° 


1.6777 


1.4740 


1 59° 

1 


2.1300 


1.2206 


89° 


5.4349 1 1.0008 



TABLES. 



The Common Logarithms ot r(«) ^o'' Values of n between 1 and 2. 



n 


i 


n 


s 

o 

i 


n 


Is 


n 


s 

S 


■ 
n 


1 














_ 




_ 


1.01 


1.9975 


1.21 


1.9617 


1.41 


1.9478 


1.61 


1.9517 


1.81 


1.9704 


1.02 


1.9951 


1.22 


r.9605 


1.42 


1.9476 


1.62 


f.9523 


1.82 


1.9717 


1.03 


1.9928 


1.23 


r.9594 


1.43 


r.9475 


1.63 


1.9529 


1.83 


1.9730 


1.04 


1.9905 


1.24 


1.9583 


1.44 


1.9473 


1.64 


1.9536 


1.84 


1.9743 


1.05 


1.9883 


1.25 


1.9573 


1.45 


1.9473 


1.65 


1.9543 


1.85 


1.9757 


V06 


L9862 


1.26 


1.9564 


1.46 


1.9472 


1.66 


1.9550 


1.86 


1.9771 


1.07 


r.9841 


1.27 


1.9554 


1.47 


1.9473 


1.67 


1.9558 


1.87 


1.9786 


1.08 


1.9821 


1.28 


1.9546 


1.48 


1.9473 


1.68 


r.9566 


1.88 


1.9800 


1.09 


1.9802 


1.29 


r.9538 


1.49 


1.9474 


1.69 


1.9575 


1.89 


1.9815 


1.10 


1.9783 


1.30 


1.9530 


1.50 


1.9475 


1.70 


r.9584 


1.90 


1.9831 


1.11 


1.9765 


1.31 


1.9523 


1.51 


1.9477 


1.71 


r.9593 


1.91 


1.9846 


1.12 


1.9748 


1.32 


1.9516 


1.52 


1.9479 


1.72 


1.9603 


1.92 


1.9862 


1.13 


1.9731 


1.33 


1.9510 


1.53 


r.9482 


1.73 


1.9613 


1.93 


1.9878 


1.14 


1.9715 


1.34 


f.9505 


1.54 


1.9485 


1.74 


1.9623 


1.94 


1.9895 


1.15 


1.9699 


1.35 


1.9500 


1.55 


1.9488 


1.75 


1.9633 


1.95 


19912 


1.16 


1.9684 


1.36 


1.9495 


1.56 


1.9492 


1.76 


r.9644 


1.96 


1.9929 


1.17 


1.9669 


1.37 


1.9491 


1.57 


1.9496 


1.77 


1.9656 


1.97 


1.9946 


1.18 


1.9655 


1.38 


1.9487 


1.58 


1.9501 


1.78 


1.9667 


1.98 


1.9964 


1.19 


1.9642 


1.39 


1.9483 


1.59 


1.9506 


1.79 


1.9679 


1.99 


1.9982 


1.20 

1 


19629 


1.40 

• 


1.9481 


1.60 


1.9511 

1 


1.80 


1.9691 


2.00 


0.0000 



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